Why is 0.3 times 0.2 equal to 0.06, and not 0.6 like some may think?
The rules for decimal multiplication and division follow directly from what we know about decimal place value and fraction arithmetic.
Example: Multiple 0.3(0.2).
Since 
and 
, multiplying 0.3(0.2) is the same as multiplying 
. We know that 
, so we know that 0.3(0.2) must equal 
which when written as a decimal is 0.06.
Note that to multiple 0.3(0.2) we only needed to multiple 3(2) = 6 and then to place the decimal point two place values from the right (0.06) since tenths time tenths equals hundredths. This motivates the process for answering the question:
Why Does the General Algorithm for Multiplying Decimals Work?
You may remember that the general rule for multiplying decimals is to multiply the digits as if they were whole numbers and then place the decimal point in the answer based on the number of "decimal places" in the factors being multiplied.
Example: Multiply 19.6(0.073)
Compare this to the common fraction and exponent form:
This rule is just a short cut description of what happens when we multiply the two decimals in their fraction forms. When a decimal value is translated into fraction form using place value, the role of the decimal point is replaced by the power of ten (place value) used in the denominator.
Examples:
This means that when two decimal values are multiplied, the product will be the product of these corresponding fractions.
Example:
And that in turn, means that the numerator of the product is the product of the decimal digits as whole numbers without regard to the decimal point. This explains why we can begin multiplying decimals by multiplying the values like we would whole numbers, without regard to the decimal point. It is because, in the fraction form, the decimal point no longer determines the place value.
So Why Can We Just Count up the Decimal Places for the Decimal Point in the Answer?
In the decimal form, the number of decimal places determines the place value and with it, the power of 10 for that place value.
Example: 0.3 is "three tenths" and written as a fraction is 
.
One decimal place is "tenths" which means that 101 is in the denominator.
Two decimal places is "hundredths" which means that 102 is in the denominator.
When we studied simplifying exponential expressions we learned that bx · by = b(x+y). Applying that to our decimal multiplication situation, we can think of the denominator of the first factor is 10x and the denominator of the second factor is 10y. Then the denominator of the product must be 10x · 10y =10x+y. It follows that the total number of decimal places in the two factors will be the equal to the total number of decimal places in the product. So when we multiply two decimal values, we can multiply their digits as if they were whole numbers (they would be in the fraction numerators) and the decimal point in the answer will make the same number of decimal places in the answer as the total of decimal places in the values being multiplied.
Lynn bought 3.7 pounds of hamburger at a price of $2.58 per pound. How much did the hamburger cost Lynn.
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Date last modified: June 30, 2010.
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